The development of nuclear structure physics is driven by the discovery of new phenomena and the need to understand them. One of such challenges is nuclear hyperdeformation (HD); the state of nucleus characterized by extreme shape elongation. Theoretical investigations of HD at high spin were carried out only in the framework of the macroscopic+microscopic method so far. Our study within the cranked relativistic mean field (CRMF) theory, which covers the Z=40-58 part of nuclear chart, is a first systematic investigation of the hyperdeformation in a fully self-consistent theory. It is centered around the following questions: (i) what are the best regions for the observation of nuclear HD, (ii) which features of nuclear many-body system may prevent the observation of discrete HD bands, (iii) what are the general features of the HD rotational bands, and (iv) which methods can be used for configuration assignment of the HD bands.

Current experimental efforts are concentrated in the valley of beta-stability where CRMF calculations show that the HD states become yrast only at very high spin of I ~ 80 hbar. On the contrary, the calculations show that by going towards proton-drip line one can lower these spins by 10 hbar and more: this can make the observation of the discrete HD bands more feasible. The small size of the HD shell gaps and the softness of the HD minimum leads to high density of the HD bands in many cases. This will most likely prevent the observation of discrete HD bands since the feeding intensity will be distributed among many bands. On the other hand, it will favor the observation of rotational patterns in the form of ridge-structures in three-dimensional rotational mapped spectra: this represents another way of study of HD. Our calculations suggest ^{111} I and Cd isotopes as the candidates for a search of discrete HD bands (Cd isotopes) or the bands with properties close to HD (^{111} I). A centrifugal stretching of the HD shapes with increasing rotational frequency, leading to an increase of transition quadrupole moments as well as kinematic and dynamic moments of inertia, is a general feature of the HD bands. The methods of configuration assignment for the HD bands will also be discussed.